Strong-coupling theory of magnetic-exciton-mediated superconductivity in UPd$_2$Al$_3$

a r X i v :c o n d -m a t /0401520v 1 [c o n d -m a t .s t r -e l ] 26 J a n 2004Strong-coupling theory of magnetic-exciton-mediated superconductivity in UPd 2Al 3P.McHaleMax-Planck-Institut f¨u r Physik komplexer Systeme,N¨o thnitzer Str.38,01187Dresden,GermanyP.ThalmeierMax-Planck-Institut f¨u r Chemische Physik fester Stoffe,N¨o thnitzer Str.40,01187Dresden,GermanyP.Fulde Max-Planck-Institut f¨u r Physik komplexer Systeme,N¨o thnitzer Str.38,01187Dresden,Germany (Dated:February 2,2008)There is compelling evidence from inelastic-neutron-scattering and tunneling experiments that the heavy-fermion superconductor UPd 2Al 3can be understood as a dual system consisting of magnetic excitons,arising from crystal-field-split U 4+levels,coupled to delocalised f-electrons.We have computed the superconducting transition temperature and the mass renormalisation arising from a dual model with maximal spin anisotropy using a strong-coupling approach.We find an instability to two possible opposite-spin-pairing states with even-or odd-parity gap functions.Each has a line node perpendicular to the c-direction,in agreement with NMR relaxation-rate,specific-heat and thermal-conductivity measurements.In addition,both have total spin component S z =0,compatible with the observation of a pronounced Knight shift and H c 2Pauli limiting.For parameter values appropriate to UPd 2Al 3,we determine the dependence of the superconducting transition temperature T c on a phenomenological coupling constant g and we investigate the associated mass enhancement and its anisotropy.PACS numbers:74.20.Mn,74.20.Rp I.INTRODUCTION Heavy-fermion superconductors are complicated materials which possess a fascinating and rich variety of physical properties and which have stimulated the creation of an equally diverse range of theories in an effort to understand them.These materials are dominated by strong electronic correlations giving rise to magnetic and superconducting instabilities.Accordingly,most theories ignore phononic degrees of freedom.One of the most extensively studied models of heavy-fermion superconductivity is a phenomenological model based on the exchange of spin fluctuations between heavy quasiparticles.The latter originate in periodic resonant scattering of conduction electrons close to the Fermi level as described within the Kondo lattice model of Ce compounds 1,2.Pairing via spin fluctuations can be considered as a one-component model to the extent that the spin fluctuations originate in the system of heavy quasiparticles and any interaction between conduction electrons and more localised electrons,or conduction electrons and phonons,is neglected.In spite of its simplicity this model has proved to be very useful in our attempts to understand heavy-fermion compounds on the border of magnetic long-range order,such as CePd 2Si 23,CeIn 33and CeRhIn 54.Its success probably derives from the fact that the spin-fluctuation-induced effective interaction tends to dominate all other channels of interaction when a material is tuned close to the border of magnetism by doping or,as in the Ce compounds mentioned above,by applying pressure.The spin-fluctuation mechanism in the Kondo lattice may also be appropriate for the Ce-based heavy-fermion superconductors at ambient pressure,CeCu 2Si 2and CeCoIn 54,5.It has become increasingly evident that the Kondo-lattice model is,however,not adequate in the case of heavy-quasiparticle formation in some Uranium heavy-fermion compounds 6,7,where 5f electrons are partly localised and partly itinerant.The former occupy crystalline-electric-field (CEF)split 5f 2states.The latter,more itinerant,5f electrons have a strongly enhanced effective mass due to a coupling to virtual excitations between CEF states.Thisnotion of heavy-quasiparticle formation is especially appropriate for UPd 2Al 3and it is the starting point of our theoretical model for heavy-fermion superconductivity in this compound.Specifically,we consider a novel effective pairing mechanism which is based on the virtual exchange of collective CEF excitations known as ‘magnetic excitons’.They are propagating bosonic modes in contrast to the overdamped modes which give rise to pairing in the itinerant spin-fluctuation model.The interplay of conduction electrons and CEF excitations is a well studied subject.Initially,research was focused in two areas 8:transport anomalies and superconducting pair breaking or enhancement due to conduction-electron scat-tering from dilute CEF-split impurities;and periodic lattices of CEF ions interacting with one another via conduction-electron polarisation.In the latter case attention was focused on collective effects within the CEF system due to the RKKY interactions mediated by itinerant ter,White and Fulde 9showed that the inverse effect,viz.conduction-electron mass enhancement via virtual emission and absorption of magnetic excitons,is also important.They demonstrated that this mechanism explains the enhanced effective electron masses in praseodymium metal.It is then natural to ask whether the exchange of magnetic excitons between quasiparticles can also mediate superconductivity.Until recently,no theoretical work had been carried out to answer this question,mainly for two reasons.Firstly,when the effects on superconductivity of paramagnetic impurities with CEF-split energy levels were studied in detail 10,it was found that for rare-earth ions the pair-breaking transition matrix elements usually dominate the pair-enhancing matrix elements.Secondly,no good example of a superconducting compound with this dual nature,i.e.,magnetic excitons arising from localised f electrons coupled to delocalised f electrons,was known.Recently,however,evidence has been accumulating that the heavy-fermion compound UPd 2Al 3(T N =14.3K,T c =1.8K,see Ref.11)is the first example where such a mechanism is responsible for superconductivity.Since this mechanism is pair breaking in the s-wave channel (as was already known from the impurity models mentioned above),the superconducting gap function must change sign as a function of k (i.e.,the gap must have a node)as is the case with spin-fluctuation-mediated superconductivity.An early indication that UPd 2Al 3is a localised-delocalised f system came from magnetic-susceptibility measure-ments.Grauel et al 12measured the dc magnetic susceptibility in UPd 2Al 3and argued that the anisotropy which they observed arose from a tetravalent configuration of the Uranium ions,i.e.U 4+(5f 2).Since then Knight-shift 13and optical 14measurements have supported the notion that UPd 2Al 3contains both localised and delocalised f electrons.However,the most direct evidence in favour of the dual-system hypothesis comes from inelastic neutron scattering (INS)and tunnelling experiments.A dispersive crystal-field excitation (magnetic exciton)was observed by Mason and Aeppli 15in INS experiments within the AF ter higher-resolution experiments 16–18revealed a resonance feature which appears in the INS spectrum upon entering the superconducting state.This result demonstrates that a strong interaction exists between the localised and delocalised components of the f-electron system.Pioneering tunnelling experiments 19performed on UPd 2Al 3-AlO x -Pb tunnelling junctions allowed experimenters to view,for the first time,the tunnelling density of states (DOS)of a heavy-fermion superconductor.Strong-coupling features appear in the DOS around 1meV close to the superconducting gap energy reinforcing the view that the exchanged bosons are the magnetic excitons.Taken together,INS and tunnelling experiments lead Sato and co-workers 20to the conclusion that superconductivity arises in this material from an effective interaction between itinerant electrons mediated by magnetic ing a model two-component Hamiltonian they were able to explain qualitative features of the INS scattering spectrum and the superconducting tunnelling spectrum.Subsequently,the origin of the magnetic excitons and their global dispersion,as measured in Ref.15,was investigated in more detail ing an xy-type interaction,where only the σ±components of the conduction-electron spin couples to the magnetic excitons,the effective non-retarded pair potential was derived.The gap equations were solved within a weak-coupling approach demonstrating that the highest T c is obtained by an odd-parity state.In this model,however,the structure of the pairing amplitudes in spin space is complicated,making it unsuitable for going beyond the non-retarded approximation.In this paper,therefore,we investigate an alternative model with a simplified interaction between localised and itinerant 5f electrons.This interaction is of the Ising type,i.e.,only the σz component of the conduction-electron spin can scatter magnetic excitons.In this case,the gap equations naturally divide into those for equal-and opposite-spin pairing,in contrast to the usual ‘singlet’and ‘triplet’classification that arises in spin-rotation-symmetric models.In the present work we treat this simplified dual model for UPd 2Al 3in a more sophisticated strong-coupling ap-proach using a mapping to an electron-boson Hamiltonian and solving the ´Eliashberg equations for the frequency-and momentum-dependent self-energy and gap functions.The level scheme which we use here,and which gives rise to the Ising-type interaction,is quite realistic:recent band-structure calculations based on this level scheme account well for experimental de Haas-van Alphen frequencies in UPd 2Al 37.We demonstrate that our model can yield a super-conducting transition temperature and a mass renormalisation which are consistent with experiment for reasonable values of the coupling constant.II.MODELBand structure calculations based on the supposition that two of the three Uranium 5f-electrons are localised reproduce the observed de Haas-van Alphen frequencies in UPd 2Al 3very well 7.These calculations suggest a level scheme for the localised U 5f states which we adopt here.According to the jj-coupling scheme the U 4+(5f 2)ions have total angular momentum J =4.The twofold degeneracy of the ionic ground state is lifted by a crystalline electric field.We consider only the excitation between the non-degenerate ground state |Γ4 =12(|J z =3 +|J z =−3 ).The CEF energy splitting is of the order ∆≃6meV,asobtained from fitting the magnetic-exciton dispersion obtained from INS 15measurements to theoretical results 21.Although such a model can give rise to induced magnetism via mixing of the states |Γ3 and |Γ4 ,we take theview here that the underlying antiferromagnetic(AF)order is not an important consideration in the description ofheavy-fermion superconductivity in UPd2Al3.AF order mainly leads to a folding down of the conduction bands into the AF Brillouin zone.Also,the magnetic-exciton dispersion in the ordered phase is not appreciably different fromthat in the paramagnetic phase,due to the fact that T N≪∆21.Furthermore,in mean-field theory,the AF order parameter will only slightly modify the superconducting T c due to a reconstruction of the conduction-electron statesclose to the AF Bragg planes.Accordingly we consider a three-dimensional lattice of localised5f2CEF states and itinerant f-electrons.In thesubspace{|Γ3 ,|Γ4 }we may write the CEF part of the Hamiltonian asH CEF=∆ i S iz,(1)where S denotes a pseudospin(S=12 and energy−∆/2and the excited CEF state as having S z=121√2λq αq+α†−q e i qr;λ2q=∆ψβ(r′,τ′) (7)D(r−r′,τ−τ′)=− Tφ(r,τ)φ(r′,τ′) .(8) Note thatφ(r)∼S x(r),as can be seen immediately by comparing the terms∼I in Eqs.(3)and(5),and so D is essentially a pseudospin susceptibility.Furthermore,as in the theory of phonon-mediated superconductivity,φ(r)is real,φ(r)=φ†(r),and commutes with itself,[φ(r),φ(r′)]=0,conditions which allow us to use Wick’s Theorem.The ´Eliashberg equations for the conduction-electron self-energyΣ(p,iω)and gap functionΦ(p,iωn)which follow fromnthe Hamiltonian Eq.(5)areTΣ(p,iωn)=N p′ω′n K(p−p′,iωn−iω′n)|G(p′,iω′n)|2Φ(p′,iω′n).(10)N is the total number of lattice sites.The kernel K is given byK (q ,iνn )=−I 2D 0(q ,iνn )= I 2∆ν2n +ω2q .(11)We have decided not to renormalise D 0by the interaction I between magnetic excitons and electrons.This interaction is,however,already included in D 0to the extent that we model the exciton dispersion ωq by the true experimental dispersion.A particularly noteworthy feature of this formulation is that the effective interaction is dominated by its static part and is strongly peaked in q -space at the antiferromagnetic wavevector Q =(0,0,πc )has actually been observed 17,18in UPd 2Al 3.The electron Green’s function is related to the electron self-energy via the Dysonequation G −1(p ,iωn )=iωn −(ǫp −µ)−Σ(p ,iωn ).(12)The prefactor p in Eq.(10)is the expectation value of the Ising spin-spin interaction ˆσz ˆσz in the spin part of the pair wave function |χ =|S,S z (S here should not be confused with the pseudospin introduced earlier):p = χ|ˆσz ˆσz |χ .(13)In the opposite-spin pair (OSP)states,|χ =12(|↑↓ −|↓↑ )12(|↑↓ +|↓↑ ),(14)p reduces top = ↑↓|ˆσz ˆσz |↑↓ =−1.(15)On the other hand,for each of the equal-spin pair (ESP)states,|χ =|↑↑ and |↓↓ ,p is +1.Note that spin rotational symmetry is broken in a maximal (Ising-type)way in Eq.(10).Consequently the usual classification of pairing states into an (even-parity)singlet with S =0and an (odd-parity)triplet with S =1is no longer valid.In our model,the |1,0 pair state no longer has the same energy as that of the |1,±1 pair states.The result is a different classification into an ESP doublet and two OSP singlet states belonging to irreducible representations of the D 6h group (Table I).Finally,the conduction-band filling,defined as the ratio of the number of electrons to the number of lattice sites,isn =1N z p ′z ω′n K (p z −p ′z ,iωn −iω′n )d p ′⊥N zp ′z ω′n K (p z −p ′z ,iωn −iω′n )Φ(p ′z ,iω′n ) d p ′⊥N z k z Ad k ⊥provided we can carry out the integrals along the perpendicular direction analytically.Here,N z is the number of lattice sites in the z-direction and A is the area of the (hexagonal)unit cell in the plane.We model the exciton dispersion byω(q z )=ωex [1+βcos(q z )];0<β≃1,(21)This form of the dispersion along the c-direction describes qualitatively the observed excitation branch 15.Here 2βωex ≃8meV is the overall dispersion width and (1-β)ωex ≃1meV is the exciton gap at the AF wave vector Q =(0,0,πN z p ′z ω′nK (p z −p ′z ,iωn −iω′n )G z (p ′z ,iω′n )(22)λ(T )Φ(p z ,iωn )=p TN z k z ,ωn >0ln (z ′−1)2+(z ′′)2(ωq z /ωex )2+(νn /ωex )2(25)G ′z (p ′z ,iωn )=−1(z ′)2+(z ′′)2(26)G ′′z (p ′z ,iωn )=− tan −1 1−z ′z ′′(27)M z (p ′z ,iωn )=1z ′′ −tan −1 −z ′2 12π 1TABLE I:Spin and orbital structure of the possible gap functions which are solutions of the´Eliashberg equations for the dual model of UPd2Al3used in this paper.p|χ =|S,S z D6h repres.spin pairingΦ(p z)√√2cos(p′z);α≪1(33) whereαdetermines the degree of corrugation of the FS cylinder along c.The momentum convolutions in Eqs.(22)and(23)were evaluated with the aid of a fast-Fourier-transform algorithm on a32×32lattice.The corresponding frequency sums were carried out using the renormalisation group technique of Pao and Bickers22which allows a considerable reduction of the computational effort.Between240and480Matsubara frequencies were kept at each stage of the renormalisation group procedure.The renormalisation procedure was started at a temperature T0=0.01ǫ⊥and the frequency sum cut-offused wasΩc≈15ǫ⊥.The renormalisation procedure restricts us to discrete temperatures so that the point at which the condition in Eq.(29)is met must be determined by interpolation.The discrete temperatures were sufficiently close that a linear interpolation was adequate.III.RESULTS AND DISCUSSIONScattering by isolated impurities with CEF excitations is usually pair breaking in the s-wave channel8because the dipolar exchange interactions which break the singlet state are stronger than spin-conserving quadrupolar interactions between conduction electrons and CEF states.The s-wave state(whereΦ(p z)is nodeless)is also not favourable in our case of a periodic lattice of partly localised5f-electrons which has dispersive CEF excitations.One can most clearly see this in the following way.In the singlet channel p=−1.Because the kernel K(q,iνn)is strongly peaked at q z=πc)Φ(p z−πc )is a positive number.Therefore afinite gapΦ(p z,iπT)must change sign on translation throughQ z=πc )=−Φ(p z),which excludes an s-wave gap function.However a non-s-wave superconducting state with nodes may take advantage of the exciton dispersion and become stable.From numerical solutions of the linearised gap equations for our model wefind that the instability with the highest T c is accidentally doubly degenerate.The corresponding gap functions transform as the even-and odd-parity OSP states,cos(cp z)and sin(cp z).The former is the usual singlet state;the latter is the S z=0part of the triplet which would appear in a theory with full spin rotational symmetry.We alsofind an instability in the ESP channel but at a much lower temperature.The corresponding pairing symmetry isΦ(p z)∼sin(2cp z).We summarise our results in Table I.The symmetries of the orbital pair wavefunctions can be understood in the following way.By comparing our self-energy equations with those obtained from a four-fermion interaction we see that the effective interaction between quasiparticles in our theory isγ2γ4|ˆV|γ1γ3 =I2D0σzγ1γ2σzγ3γ4,(35) or equivalentlyˆV=I2D0ˆσzˆσz.(36) Hence the pairing interaction in some|χ channel isVχ(q,iνn)= χ|ˆσzˆσz|χ I2D0(q,iνn).(37) We note that our equations go beyond previous calculations6,21in that our formulation includes the full momentum and frequency dependence of the effective interaction Vχ(q,iνn).In the OSP channel the static interaction isV(q z)=−I2D0(q z)=I2∆/2c .The corresponding interaction in real space is therefore attractive when the quasiparti-cles are seperated by a lattice spacing in the c-direction.Now cos(cp z)and sin(cp z)are peaked with equal amplitude in real space at z=c.Hence they are equally well suited to take advantage of the attractive part of the interaction and have equal superconducting transition temperatures.The ESP interaction is the negative of the OSP interaction and so itfirst becomes attractive at z=2c in real space.The only odd-parity wavefunction which is peaked in real space at this position is sin(2cp z).Ultimately,the degeneracy of the OSP states is a peculiarity of the particular CEF level scheme we adopt for the localised5f states and of the approximate form of their exciton dispersion which we assume(viz.strongly wavevector dependent only in the z-direction).Both OSP states are,in fact,compatible with present experimental evidence. OSP states,regardless of the value of total spin angular momentum S,give rise to a reduction in the paramagnetic susceptibility on entering the superconducting state.This reduction comes about because oppositely paired spins have no magnetisation.Furthermore,such a reduction in the Pauli susceptibility leads to an upper criticalfield H c2 which,in some cases,can be smaller than that due to the Meissner effect.This effect is called Pauli limiting23.A pronounced Knight-shift reduction at Pd sites below T c(Refs.13,24)and H c2Pauli-limiting25have both been observed in UPd2Al3.Note also that both OSP gap functions have node lines perpendicular to the c-direction.In the even-parity case they are located at k z=±12N(µ)ǫ⊥12(λ1λ2)2λ3,(40)FIG.1:The dependence of the superconducting transition temperature T c on the coupling constant g[Eq.(30)].The band filling n is0.6.FIG.2:The dependence of the mass renormalisation m∗/m b on the coupling constant g[Eq.(30)].The bandfilling n is0.6 as in Fig.1.m∗min and m∗max are the extremal effective masses over all values of q z.m∗min occurs at q z=0;m∗max occurs at q z=πwhere the dimensionless constantsλi are defined byλ1=IN(µ)(41)λ2=∆/ωex(42)1λ3=)c in UPd2Al3.Solutions of the´Eliashberg equations show that the model favours a superconducting instability for the even-or odd-parity OSP states.Each state has line nodes perpendicular to the c-direction and total z-component of spin S z=0,characteristics which are compatible with measurements of UPd2Al3.Wefind that a superconducting transition temperature and mass renormalisation,each in good agreement with their experimental values in UPd2Al3, can be obtained using reasonable values of the parameters in the theory.Taken together,these results strengthen the argument in favour of a magnetic-exciton system in UPd2Al3.The evidence for dual f-systems in other Uranium compounds does not weigh as heavily as it does in the case of UPd2Al3.In UPt3the number of itinerant f-electrons remains controversial.However band-structure calculations based on the assumption that two of the three5f-electrons are localised reproduce the observed de Haas-van Alphen frequencies as well as the anisotropic heavy electron masses in this compound very well7.Recently unconventional superconductivity with split-T c superconductivity has been discovered in the Pr-based skutterudite PrOs4Sb1233,34.Thermodynamic and transport measurements suggest that the observed heavy-fermion state arises from the interaction of electric quadrupole moments of the CEF-split4f2states of Pr3+with the conduc-tion electrons33.The ground state is probably a singlet and thefirst excited state is probably a triplet,approximately 0.5meV higher,with large off-diagonal quadrupolar matrix elements connecting the two states.The largeness of these matrix elements,together with the very small singlet-triplet splitting,strongly suggests an effective mass renormalisa-tion via virtual CEF excitations.Finally,experiments reveal that the superconducting order parameter is anisotropic, making this compound another candidate for CEF-exciton-mediated superconductivity,but this time of quadrupolar nature.UPd2Al3provides a strong motivation for the study of dual f-electron systems such as the one discussed in this work. We have considered a dual model in which all parameters arefixed except the coupling constant.It will be interesting to investigate the dependence of the transition temperature on conduction-bandfilling and on the form of the exciton dispersion,and to look for a mechanism that can discriminate between the even-and odd-parity superconducting OSP states.AcknowledgmentsP.M.would like to thank P.Monthoux for valuable discussions.P.T.would like to thank K.Maki,Y.Matsuda, K.Izawa,T.Watanabe and G.Varelogiannis for helpful discussions.The authors acknowledge discussions with G. Zwicknagl and A.Yaresko.APPENDIXIn this appendix we sketch the derivation of Eq.(5).Our starting point is the set of Holstein-Primakofftransfor-mations(valid for T≪∆),S z=−12(a+a†)(A.2)Inserting these expressions into the second and third sums in Eq.(3)and Fourier transforming,we arrive at the following form,q W1(q)a†q a q−W2(q)2N kq c†kασzαβc k+qβ(a q+a†−q),(A.3)whereW1(q)=∆−W2(q).(A.4) Thefirst sum can be diagonalised35,while at the same time introducing only a factorλq in the second sum.To do this we use the Bogoliubov transformationαkα†−k = u k v k v k u k a k a†−k,(A.5)where u k=cosh(θk)and v k=sinh(θk).Ifθk is chosen such that−W2(k)/W1(k)=tanh(2θk),(A.6) then(ignoring constant additive terms)thefirst sum in Eq.(A.3)takes the form,qωq(α†qαq+1/2)(A.7) where the dispersionωq satisfiesω2q=W21(q)−W22(q).(A.8) Finally,it is easy to show thata q+a†−q=λq(αq+α†−q),(A.9) whereλq=cosh(θq)−sinh(θq).(A.10) Using Eqs.(A.4)and(A.6),together with the identity∆−2W2(q)=ω2qωq.(A.12) The interaction term in Eq.(5)follows immediately.1A.C.Hewson,The Kondo Problem to Heavy Fermions(Cambridge University Press,1993).11 2P.Fulde,J.Keller,and G.Zwicknagl,in Solid State Physics(Academic Press,San Diego,1988),vol.41,p.1.3N.D.Mathur,F.M.Grosche,S.R.Julian,I.R.Walker,D.M.Freye,R.K.W.Haselwimmer,and G.G.Lonzarich,Nature 394,39(1998).4N.J.Curro,J.L.Sarrao,J.D.Thompson,P.G.Pagliuso,ˇS.Kos,A.Abanov,and D.Pines,Phys.Rev.Lett.90,227202 (2003).5C.Petrovic,P.G.Pagliuso,M.F.Hundley,R.Movshovich,J.L.Sarrao,J.D.Thompson,Z.Fisk,and P.Monthoux,J.Phys.Condens.Matter13,L337(2001).6G.Zwicknagl,A.N.Yaresko,and P.Fulde,Phys.Rev.B65,081103(2002).7G.Zwicknagl,A.Yaresko,and P.Fulde,Phys.Rev.B68,052508(2003).8P.Fulde and I.Peschel,Adv.Phys.21,1(1972).9R.M.White and P.Fulde,Phys.Rev.Lett.47,1540(1981).10P.Fulde,L.L.Hirst,and A.Luther,Z.Phys.230,155(1970).11C.Geibel,C.Schank,S.Thies,H.Kitazawa,C.D.Bredl,A.B¨o hm,M.Rau,A.Grauel,R.Caspary,R.Helfrich,et al.,Z.Phys.B84,1(1991).12A.Grauel,A.B¨o hm,H.Fischer,C.Geibel,R.K¨o hler,R.Modler,C.Schank,F.Steglich,and G.Weber,Phys.Rev.B46, 5818(1992).13R.Feyerherm,A.Amato,F.N.Gygax,A.Schenck,C.Geibel,F.Steglich,N.Sato,and T.Komatsubara,Phys.Rev.Lett.73,1849(1994).14M.Dressel,N.Kasper,K.Pethukhov,B.Gorshunov,G.Gr¨u ner,M.Huth,and H.Adrian,Phys.Rev.Lett.88,186404 (2002).15T.E.Mason and G.Aeppli,Matematisk-fysiske Meddelelser45,231(1997).16N.Sato,N.Aso,nder,B.Roessli,T.Komatsubara,and Y.Endoh,J.Phys.Soc.Japan66,1884(1997).17N.Bernhoeft,N.Sato,B.Roessli,N.Aso,A.Hiess,nder,Y.Endoh,and T.Komatsubara,Phys.Rev.Lett.81, 4244(1998).18N.Bernhoeft,Eur.Phys.J.B13,685(2000).19M.Jourdan,M.Huth,and H.Adrian,Nature398,47(1999).20N.K.Sato,N.Aso,K.Miyake,R.Shiina,P.Thalmeier,G.Varelogiannis,C.Geibel,F.Steglich,P.Fulde,and T.Komat-subara,Nature410,340(2001).21P.Thalmeier,Eur.Phys.J.B27,29(2002).22C.-H.Pao and N.E.Bickers,Phys.Rev.B49,1586(1994).23A.M.Clogston,Phys.Rev.Lett.9,266(1962).24H.Tou,K.Ishida,and Y.Kitaoka,cond-mat/0308562(2003).25J.Hessert,M.Huth,M.Jourdan,H.Adrian,C.T.Rieck,and K.Scharnberg,Physica B230-232,373(1997).26H.Tou,Y.Kitaoka,K.Asayama,C.Geibel,C.Schank,and F.Steglich,Phys.Soc.Jpn.64,725(1995).27K.Matsuda,Y.Kohori,and T.Kohara,Phys.Rev.B55,15223(1997).28N.Sato,unpublished.29M.Hiroi,M.Sera,N.Kobayashi,Y.Haga,E.Yamamoto,and Y.Onuki,J.Phys.Soc.Jpn.66,1595(1997).30P.Thalmeier and K.Maki,Europhys.Lett.58,119(2002).31P.F.J.Jensen,Phys.Rev.B27,4085(1983).32Y.Inada,H.Yamagami,Y.Haga,K.Sakurai,Y.Tokiwa,T.Honma,E.Yamamoto,Y.¯Onuki,and T.Yanagisawa,J.Phys.Soc.Japan68,3643(1999).33E.D.Bauer,N.A.Frederick,P.-C.Ho,V.S.Zapf,and M.B.Maple,Phys.Rev.B65,100506(2002).34R.Vollmer,A.Faisst,C.Pfleiderer,H.v.L¨o hneysen,E.D.Bauer,P.C.Ho,and M.B.Maple,Phys.Rev.Lett.90,57001 (2003).35B.Grover,Phys.Rev.140,A1944(1965).。